Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles

We consider cosmological models of Bianchi type. In particular, we are interested in the α-limit dynamics near the Kasner circle of equilibria for Bianchi classes VIII and IX. They correspond to cosmological models close to the big-bang singularity.     

Bianchi cosmologies with anisotropic matter: Locally rotationally symmetric models

The dynamics of cosmological models with isotropic matter sources (perfect fluids) is extensively studied in the literature; in comparison, the dynamics of cosmological models with anisotropic matter sources is not. In this paper we consider spatially homogeneous locally rotationally symmetric solutions of the Einstein equations with a large class of anisotropic matter models including collisionless matter (Vlasov), elastic matter, and magnetic fields. The dynamics of models of Bianchi types I, II, and IX are completely described; the two most striking results are the following. (i) There exist matter models, compatible with the standard energy conditions, such that solutions of Bianchi type IX (closed cosmologies) need not necessarily recollapse; there is an open set of forever expanding solutions. (ii) Generic type IX solutions associated with a matter model like Vlasov matter exhibit oscillatory behavior toward the initial singularity. This behavior differs significantly from that of vacuum/perfect fluid cosmologies; hence “matter matters”. Finally, we indicate that our methods can probably be extended to treat a number of open problems—in particular, the dynamics of Bianchi type VIII and Kantowski-Sachs solutions.     

Semiclassical and quantum behavior of the Mixmaster model in the polymer approach for the isotropic Misner variable

We analyze the semi-classical and quantum behavior of the Bianchi IX Universe in the Polymer Quantum Mechanics framework, applied to the isotropic Misner variable, linked to the space volume of the model. The study is performed both in the Hamiltonian and field equations approaches, leading to the remarkable result of a still singular and chaotic cosmology, whose Poincaré return map asymptotically overlaps the standard Belinskii–Khalatnikov–Lifshitz one. In the quantum sector, we reproduce the original analysis due to Misner, within the revised Polymer approach and we arrive to demonstrate that the quantum numbers of the point-Universe still remain constants of motion. This issue confirms the possibility to have quasi-classical states up to the initial singularity. The present study clearly demonstrates that the asymptotic behavior of the Bianchi IX Universe towards the singularity is not significantly affected by the Polymer reformulation of the spatial volume dynamics both on a pure quantum and a semiclassical level.     

Quantum suppression of the generic chaotic behavior close to cosmological singularities

In classical general relativity, the generic approach to the initial singularity is very complicated as exemplified by the chaos of the Bianchi IX model which displays the generic local evolution close to a singularity. Quantum gravity effects can potentially change the behavior and lead to a simpler initial state. This is verified here in the context of loop quantum gravity, using methods of loop quantum cosmology: The chaotic behavior stops once quantum effects become important. This is consistent with the discrete structure of space predicted by loop quantum gravity.     

The Mixmaster Universe in a generalized uncertainty principle framework

The Bianchi IX cosmological model is analyzed in a generalized uncertainty principle framework. The Arnowitt–Deser–Misner reduction of the dynamics is performed and a time-coordinate, namely the volume of the Universe, naturally arises. Such a variable is treated in the ordinary way while the anisotropies (the physical degrees of freedom) are described by a deformed Heisenberg algebra. The analysis of the model (passing through Bianchi I and II) is performed at classical level by studying the modifications induced on the symplectic geometry by the deformed algebra. We show that the Universe cannot isotropize because of the deformed Kasner dynamics, the triangular allowed domain is asymptotically stationary with respect to the particle (Universe) and its bounces against the walls are not interrupted by the deformed effects. Furthermore, no reflection law can be in general obtained since the Bianchi II model is no longer analytically integrable. This way, the deformed Mixmaster Universe can be still considered as a chaotic system.     

Asymptotic Regimes of Magnetic Bianchi Cosmologies

We consider the asymptotic dynamics of the Einstein-Maxwell field equations for the class of non-tilted Bianchi cosmologies with a barotropic perfect fluid and a pure homogeneous source-free magnetic field, with emphasis on models of Bianchi type VII₀, which have not been previously studied. Using the orthonormal frame formalism and Hubble-normalized variables, we show that, as is the case for the previously studied class A magnetic Bianchi models, the magnetic Bianchi VII₀ cosmologies also exhibit an oscillatory approach to the initial singularity. However, in contrast to the other magnetic Bianchi models, we rigorously establish that typical magnetic Bianchi VII₀ cosmologies exhibit the phenomena of asymptotic self-similarity breaking and Weyl curvature dominance in the late-time regime.     

Asymptotic Self-similarity Breaking in Cosmology

We discuss the role of self-similarity in the evolution of cosmological models. The simplest model, the flat Friedmann–Lematre universe is exactly self-similar. On the other hand, the open Friedmann–Lematre universe and the anisotropic Bianchi I universes, are not exactly self-similar, but are asymptotically self-similar, both near the initial singularity and at late times. In general, however, cosmological models are not asymptotically self-similar, and our goal is to describe the different mechanisms that lead to asymptotic self-similarity breaking in Bianchi universes. The discussion will also serve to give an overview of our current understanding of the dynamics of Bianchi universes.     

Collision-free gases in Bianchi space-times

We study collision-free gases in Bianchi space-times. Spatially homogeneous distribution functions are found for all Bianchi types by supposing that the distribution functionf(x, p) is a function of the Killing vector constants of the motion only. Bianchi types I, VIII and IX only, lead to physical distributions. In types VIII and IX the average behaviour of the gas is that of a nonrotating viscous fluid. In an attempt to obtain physical spatially homogeneous distribution functions for all Bianchi types, we write the Liouville equation in a spatially homogeneous orthonormal tetrad. Furthermore, the general inhomogeneous solution of Liouville's equation in Bianchi type I is obtained, depending on constants of the motion that generalise the conserved quantities generated by Lorentz boosts in flat space-time.     

Bianchi Types II, VIII and IX String Cosmological Models with Bulk Viscosity in Brans-Dicke Theory of Gravitation

We have obtained and presented spatially homogeneous Bianchi types II, VIII and IX string cosmological models with bulk viscosity in a scalar tensor theory of gravitation proposed by Brans and Dicke (Phys. Rev. 124:925, 1961). It is observed that in case of Bianchi type-IX universe, only bulk viscous cosmological model exists. Some physical and geometrical properties of the models are also discussed.     

Perfect fluid quantum anisotropic universe: merits and challenges

The present paper deals with quantization of perfect fluid anisotropic cosmological models. Bianchi type V and IX models are discussed following Schutz’s method of expressing fluid velocities in terms of six potentials. The wave functions are found for several examples of equations of state. In one case a complete wave packet could be formed analytically. The initial singularity of a zero proper volume can be avoided in this case, but it is plagued by the usual problem of non-unitarity of anisotropic quantum cosmological models. It is seen that a particular operator ordering alleviates this problem.     

Orthogonal Bianchi cosmologies: Bounds on the evolution of some physical quantities

We obtain bounds on the time evolution of a number of physical quantities such as the red shift, microwave background temperature, Hubble function and radius of the universe for any orthogonal Bianchi model except type IX. We also consider the propagation of primordial magnetic fields in such models.     

The effect of the electromagnetic fields on the evolution of homogeneous cosmological models near the singularity

Homogeneous cosmological models of all Bianchi types with a general (nondiagonal) metric are considered near the singularity. The spaces are filled with an arbitrary sourceless six-component electromagnetic field (EMF) and a perfect fluid at rest. It is shown that in the general case the models of types VI₀, VII₀, VIII, and IX have an oscillatory regime. The models of all the other types have Kasner asymptotics. The main result of the present paper is the derivation of the law of rotation of the Kasner axes.     

Fluctuations produced by the cosmological constant in the empty Bianchi type-IX model

We analyze the fluctuations of the gravitational field, produced by the cosmological term, in the empty Bianchi type-IX cosmological model. It is shown that such fluctuations do not change the process of alternation of Kasner eras very near to the singularity, and that a positive cosmological constant leads to an asymptotic isotropization of the model.     

Bianchi I cosmology in the presence of a causally regularized viscous fluid

We analyze the dynamics of a Bianchi I cosmology in the presence of a viscous fluid, causally regularized according to the Lichnerowicz approach. We show how the effect induced by shear viscosity is still able to produce a matter creation phenomenon, meaning that also in the regularized theory we address, the Universe is emerging from a singularity with a vanishing energy density value. We discuss the structure of the singularity in the isotropic limit, when bulk viscosity is the only retained contribution. We see that, as far as viscosity is not a dominant effect, the dynamics of the isotropic Universe possesses the usual non-viscous power-law behaviour but in correspondence to an effective equation of state, depending on the bulk viscosity coefficient. Finally, we show that, in the limit of a strong non-thermodynamical equilibrium of the Universe mimicked by a dominant contribution of the effective viscous pressure, a power-law inflation behaviour of the Universe appears, the cosmological horizons are removed and a significant amount of entropy is produced.     

Bianchi A Cosmological Models as the Simplest Dynamical Systems in R4

The behaviour of the vacuum non-tilted Bianchimodels of class A is studied in terms of dynamicalsystems theory. We introduce phase variables in whichthe Hamiltonian constraint is solved algebraically. It is shown that in these variables BianchiVIII and Bianchi IX models assume the form of afour-dimensional autonomous system with a polynomialvector field defined on the phase space, whereas Bianchi I and Bianchi II world models can be presentedas a one- and two-dimensional system, respectively. TheBianchi VI₀ and Bianchi VII₀ worldmodels are represented as a three-dimensional dynamicalsystem.     

On the initial singularity in the scalar-tensor anisotropic cosmology

In connection with the problem of the initial singularity in the scalar-tensor anisotropic cosmology of Jordan-Brans-Dicke, the dynamics of homogeneous models of Bianchi type I is examined on the basis of the general analytic solutions in vacuo and in the presence of gravitating matter with state equations P=n? (0 ? n ? 1). It is shown that the scalar homogeneous ?-field, as an effective source of the V₄geometry, has to influence essentially the dynamics of the early anisotropic stage of the Universe's expansion, and significantly modifies the character of the initial singularity. At negative ω (ω < ?6), the sourceless scalar ?-field may remove the singularity and provide regular ‘bouncing’ in the models with matter (P ? ?/3) if it prevails over the tensor anisotropic mode of the vacuum gravitational field.     

Numerical solutions for symmetric Bianchi type IX universes

We find numerical solutions for symmetic Bianchi type IX universes. These numerical solutions coincide almost exactly with diagrammatic solutions found previously.     

Inflationary Model of the Universe with a Type IX Metric and Possibility of Its Quantum Birth

Russian Physics Journal - Within the framework of the general theory of relativity, an anisotropic cosmological model with expansion and rotation with a Bianchi type IX metric is constructed. The...     

Quantum fluctuations near the classical space-time singularity

The method of path integration is used to study the effects of quantum fluctuations in the space-time geometry near the classical singularity of general relativity. It is shown that in certain special cases explicit Feynman propagators can be constructed which enable us to evaluate these fluctuationsquantitatively. The cases discussed are (i) the gravitational collapse of a uniform dust ball, (ii) the Friedmann cosmologies, (iii) the axisymmetric Bianchi type I cosmological model, and (iv) the general anisotropic Bianchi type I cosmological model. In all cases discussed here the quantum uncertainty grows to infinity as the classical space-time singularity is approached. In this wider regime of quantum gravitation nonsingular solutions can occur with finite probabilities.     

Anisotropic Bianchi VIII/IX cosmological models with matter and electromagnetic fields

We present a class of cosmological solutions of Einstein equations, with matter and electromagnetic field, which are anisotropic and spatially homogeneous of Bianchi types VIII and IX.